View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server COLO-HEP-396 Structure of the QCD Vacuum As Seen By Lattice Simulations T. DeGrand, Anna Hasenfratz, Tam´as Kov´acs Physics Department, University of Colorado, Boulder, CO 80309 USA Abstract This talk is a review of our studies of instantons and their properties as seen in our lattice simulations of SU(2) gauge theory. We have measured the topological susceptibility and the size distribution of instantons in the QCD vacuum. We have also investigated the properties of quarks moving in instanton background field configurations, where the sizes and locations of the instantons are taken from simulations of the full gauge theory. By themselves, these multi-instanton configurations do not confine quarks, but they induce chiral symmetry breaking. Talk presented by T. DeGrand at the 1997 Yukawa International Seminar “Nonperturbative QCD–Structure of the QCD Vacuum” 1 Introduction What features of the QCD vacuum are responsible for confinement or for the generation of the observed structure of hadron spectroscopy? This question might, in principle, be answered by lattice simulations of non-Abelian gauge theories. We have been studying the properties of instantons from SU(2) lattice simulations. This talk is a survey of our work as presented in Refs. [1, 2, 3, 4, 5]. The study of the QCD vacuum using lattice Monte Carlo is complicated by two problems. The first one is that the dominant features of the QCD vacuum as seen in lattice simulations are short distance fluctuations (as they would be for any quantum field theory). They are basically uninteresting noise. The so- lution to this problem is to invent operators which are insensitive to the short distance behavior of the field variables. This brings the second problem: The separation of vacuum structure into short distance and long distance parts is ambiguous, and what one sees can depend strongly on the operators one uses. All direct smoothing transformations[6] distort the original lattice configuration. This makes the extraction of (continuum) short to medium distance physics, like observations of topological objects, very delicate. If any space-time symmetric smoothing transformation is repeated enough times, all the vacuum structure in any finite volume, including the simulation volume, will be washed away. Thus, it does not make sense to extrapolate one’s results to the limit of a very large number of smoothing steps. The only measurements which are physically meaningful are measurements which are extrapolated back to the original lat- tice, that is, back to zero smoothing steps. This requires careful monitoring of observables over the whole history of smoothing transformations. In QCD instantons may be responsible for breaking axial symmetry and resolving the U(1) problem. [7] The relevant observable is the topological sus- ceptibility χt, defined as the infinite volume limit of hQ2i χ = h d4xQ(x)Q(0)i = (1) t V Z where Q is the topological charge and V the space time volume. In QCD χt is a dimension-4 object with no weak coupling expansion, and a calculation of χt in physical units in the continuum requires nonperturbative techniques. In 0 the large-Nc limit the mass of the η is (probably!) related to the topological susceptibility through the Witten-Veneziano formula[8] 2 fπ 2 2 2 (mη0 + mη − 2mK )=χt. (2) 2Nf The left hand side of this equation is equal to (180 MeV)4 in the real world. 2 Based on phenomenological models, it has been argued that instantons are largely responsible for chiral symmetry breaking and the low energy hadron and glueball spectrum. [9, 10] Instanton liquid models attempt to reproduce the topological content of the QCD vacuum and conclude that hadronic correlators in the instanton liquid show all the important properties of the corresponding full QCD correlators. These models appear to capture the essence of the QCD vacuum, but their derivations involve a number of uncontrolled approximations and phenomenological parameters. Instanton physics on the lattice is as full of controversy as continuum instan- ton physics. There are presently three different ways of measuring a topological charge. The “geometric” definition[11, 12] reconstructs a fiber bundle from the lattice gauge field and identifies the second Chern number of this bundle with the topological charge. It will always give an integer, but if the configuration is sufficiently rough, it can fail catastrophically. “Algebraic” definitions[13] in- troduce some lattice discretization of Q, as a sum of closed paths of loops. The worst aspect of the algebraic definitions is that the topological charge can mix with quantum fluctuations. Finally, one can define Q through fermionic opera- tors. (For a recent example, see Ref. [14].) All these definitions have a cutoff: they cannot see instanton (like) configurations when the instanton radius is too small (typically ρ/a ' 1 − 2). It is usually not possible to specify this cutoff precisely, and it can contaminate lattice measurements. 2 Finding Instantons We have explored three different methods for extracting information about topology from lattice simulations. Each has its own strengths and weaknesses. 2.1 Inverse Blocking “Inverse blocking” does not distort the original configuration. This technique has been introduced by Hasenfratz[15] in his lectures. Imagine beginning with a set of lattice variables {V } on a lattice whose spacing is a and lattice size is L. The lattice action is a fixed-point (FP) action [16] SFP(U). The inverse blocking transformation constructs a set of fine lattice variables {U} occupying a lattice of lattice spacing a/2 and lattice size 2L, by solving the steepest-descent equation SFP(V)=min SFP(U)+κT (U, V ) , (3) {U} 3 where κT (U, V ) is the blocking kernel. Inverse blocking identifies the smoothest among the configurations that block back to the original configuration. Since for fixed point actions, topology is unchanged by inverse blocking, the strategy is to take a rough configuration (generated by Monte Carlo using a FP action), inverse block it, and measure the charge on the fine lattice, where the instantons are twice as big and charge measurement is more reliable. We did this[2] and 4 found χt = (235(10) MeV) for SU(2). This created a low-level controversy, since our number was larger than other measurements.[14, 18, 19] Inverse blocking is reliable, but it is very expensive. In four dimensions, one can only perform it once (to go from lattice spacing a to a/2). The fine configurations are still too rough to identify individual instantons. 2.2 Cycling Our smoothing mechanism for seeing instantons is called “cycling.” [3] One first performs an inverse blocking from a coarse lattice to a set of fine lattice variables by solving Eqn. 3. Now the original lattice occupies one of the 16 sublattices of the fine lattice. Next, we perform a blocking transformation to a set of coarse variables {W } based on one of the other sublattices. The delicate coherence among the fine variables is broken and the new coarse variables are strongly ordered on the shortest distance scale (as measured, for example, by the expectation value of the plaquette) while retaining all long distance physics (because they are generated by a RG blocking transformation). This is the second part of the cycling transformation Vµ(x) → Uµ(x) → Wµ(x). Cycling steps can be iterated, and a few cycling steps can reduce the plaquette to within 0.001 of its free-field value. Individual instantons can be seen after a few cycling steps. Their sizes drift with smoothing (see the next section for pictures) but the drift is small enough that we can extrapolate their properties back to zero cycling steps. We found 1/4 4 χt = 230(10) MeV for SU(2), an instanton density of about two per fm ,and a mean instanton radius of about 0.2 fm. Again, the susceptibility is higher than others’, and now the mean instanton size is smaller than other measurements, although it agrees with the predictions of the interacting instanton liquid model. [9, 10] Cycling is still quite expensive because of the inverse blocking step. This makes it hard to push to small lattice spacing and to test scaling. 4 2.3 RG Mapping Cycling produces a sequence of coarse ({V }) and fine ({U}) lattices {V }→ {U1}→{V1}→{U2}→{V2}→.... RG mapping is a technique for eliminating the inverse blocking step and generating a sequence of coarse lattices {V }→ {W1}→{W2}→... where {Wn} is an approximation to {Vn}. The idea is that, while formally the inverse blocking is non-local, for local FP actions the dependence of the fine links on the original coarse links dies away exponentially with their separation [16] and the mapping {V }→{U1(V)}→{V1(V)}can be considered local. The {Wn} lattice can be constructed from the original coarse {V } lattice as a sum of loops of {V }’s which are designed to reproduce a cycled {Vn}. We discovered that this could be done by APE-smearing: [20] from a set {V } construct a new set of links {X} by † Xµ(x)=(1−c)Vµ(x)+c/6 (Vν (x)Vµ(x +ˆν)Vµ(x+ˆν) ν=µ X6 † + Vν(x−νˆ) Vµ(x−νˆ)Vµ(x−νˆ+ˆµ)), (4) with Xµ(x) projected back onto SU(2) to generate Wµ(x). We found that a series of steps with c =0.45 was a good choice to mock up cycling. RG-mapping is very cheap, but the price is that everything which is mea- sured must be monitored carefully, and extrapolated (if necessary) back to zero mapping steps.